System Control and Rough Paths

By Terry Lyons, Zhongmin Qian. 2003

668 citations recorded by the source corpus.

Subfields: Finance

View Paper: https://www.semanticscholar.org/paper/3d2281ff533e8dcaff64608335d255fd6005faf4

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Citations in the corpus

LÉVY AREA OF WIENER PROCESSES IN BANACH SPACES (2002, background): “Therefore, by the results in [23] (see also [25]), we may extend the classical Stratonovich integration theory to Banach space valued Wiener process.”
On probability laws of solutions to differential systems driven by a fractional Brownian motion (2014, background): “We consider here B together with its iterated integrals as a rough path, and we refer to [20, 27] for further details on this concept.”
Differential Equations Driven by Rough Paths: An Approach via Discrete Approximation (2007, background): “Applications of rough path theory to stochastic equations can be found, for example, in [1, 4, 8].”
Duality on gradient estimates and Wasserstein controls (2009, background): “As mentioned there, studying a coupling of this process has a possibility of a future application to rough path theory [15, 25 ].”
Ramification of rough paths (2006, methodology): “We recall the integration theory introduced in [16] in some details since this setting is quite different from the original rough path theory developed in [26, 27].”
Regularity of laws and ergodicity of hypoelliptic SDEs driven by rough paths (2011, background): “|Xt −Xs|2 =|t−s|2H. Diﬀerential equations driven by rough paths have been studied intensely in the past decade, and this theory has now reached a certain level of maturity; we refer to the monographs [16, 32, 34] for an overview of the theory. For driving signals that are rougher than Brownian motio”
Controlled differential equations as Young integrals: A simple approach (2010, neutral): “In this subject, see [11, 16, 17, 21–23].”
Convergence rates for the full Gaussian rough paths (2011, background): “Introduction Recall that rough path theory [17, 19, 9] is a general framework that allows to establish existence, uniqueness and stability of differential equations driven by multi-dimensional continuous signals x : [0, T ]→ R of low regularity.”
Ergodicity of hypoelliptic SDEs driven by fractional Brownian motion (2009, methodology): “Regularity of solutions to equations of this kind has been well studied, pioneered by the technique of Young [34] and more recently using the theory of rough paths (see, for example, [10,21] and the references therein).”
Euler estimates for rough differential equations (2006, background): “er [0,1],G[p] Rd is called a weak geometric p-rough path (with Ho¨lder-control ω). A path in C0,1/p-Ho¨lder [0,1],G[p] Rd is called a geometric p-rough path (with Ho¨lder- control ω). Proposition 14 ([15, 17]) Let x ∈ C1/p-Ho¨lder [0,1],G[p] Rd and N > [p].Then there exists a unique lift of x to ”
Non-linear rough heat equations (2009, background): “1 Introduction The rough path theory, which was first formulated in the late 90’s by Lyons [32,33] and then reworked by various authors [18,20], offers a both elegant and efficient way”
A variation embedding theorem and applications (2005, background): “Following [9] its continuity properties are conveniently expressed in terms of the modulus ω.”
Non-degeneracy of Wiener functionals arising from rough differential equations (2007, background): “For instance, almost every joint realization of Brownian motion and L´evy’s area process is a geometric p -rough path [21, 6, 5].”
Integrability Estimates for Gaussian Rough Differential Equations (2011, background): “The definition, which is standard, can be found, for example, in [12] or [24].”
Differential equations driven by Gaussian signals (2007, background): “In conjunction with the universal limit theorem [18], i.”
Sparse arrays of signatures for online character recognition (2013, methodology): “More recently, path signatures have been used to solve diﬀerential equations driven by rough paths [11, The signature extracts enough information from the path to solve any linear diﬀerential equation and uniquely characterizes paths of ﬁnite length [5].”